LECTURE 12: DIVIDED DIFFERENCE OPERATORS
WANG, QIANG
Definition 0.1. Let X = {x1 , · · · , xn }, then si ∈ Sn acts on f ∈ Z[X] by switching
the xi and xi+1 . That is,
si . f (x1 , · · · , xi , xi+1 , · · · , xn ) = f (x1 , · · · , xi+1 , xi , · · · , xn )
For si ∈ Sn , define ∂i : Z[X] → Z[X] by
(∂i f )(x1 , · · · , xn ) =
f (x1 , · · · , xn ) − si . f (x1 , · · · , xn )
xi − xi+1
In another word, ∂i = (xi − xi+1 )−1 (1 − si ).
1. Graph of reduced words
Given w ∈ (W, S), we can define a colored graph Γ(w), called the graph of
reduced words of w, as follow. The nodes in this graph are the set of all reduced
words of w. Let u, v be two nodes of this graph, (i.e, two reduced work of w) then
u, v are connected by an edge colored (labeled) by a defining relation of (W, S) if
and only if u can be transformed to v (or vise versa) by one application of the given
defining relation.
It is clear that the defining relations s2i = e are never used in Γ(w), for all nodes
in Γ(w) are reduced words.
Example 1.1. Let the Coxeter system be (S5 ), {s1 , s2 , s3 , s4 }), and let w = [31542]
in one-line notation, then one reduced word for w could be s2 s3 s4 s3 s1 , we will just
code this word by its indices 23431. Let us show the connected component of Γ(w)
that contains this word. (As we will see later this is the whole graph)
23431JJ
JJJJ
t
tt
JJJ
t
t
JJJJ
tt
JJJ$
t
t
24341
23413
42341
23143
24314
42314
21343JJJ
24134
JJJJ
tt
t
JJJJ
tt
JJJJ
tt
JJJ
tt
21434
42134
Proposition 1.2. Given w ∈ (W, S), Γ(w) is connected.
Date: January 23, 2009.
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Proof. We prove the statement for the case of (Sn , {s1 , · · · , sn−1 }), the spirits are
the same for all other types of Coxeter group.
Induct on `(w). For the case `(w) ≤ 2 the statement clearly holds.
Now assume `(w) ≥ 3, let a = a1 · · · al and b = b1 · · · bl be two reduced words
of w. By E.P. b1 w has a reduced expression of the form a1 · · · aˆk · · · al , for some
1 ≤ k ≤ l. Thus b1 a1 · · · aˆk · · · al is another reduced expression of w.
First note that by induction, b2 · · · bl and a1 · · · aˆk · · · al are connected by a sequence of edges in Γ(b1 w), thus b1 b2 · · · bl and b1 · · · aˆk · · · al are connected by a
sequence of edges in Γ(w).
If k < l, then in Γ(wal ), a1 · · · al−1 and b1 · · · aˆk · · · al−1 are connected by a
sequence of edges, thus in Γ(w), a1 · · · al and b1 · · · aˆk · · · al are connected by a
sequence of edges. Therefore a and b are connected in Γ(w).
If k = l, then either a1 and b1 are consecutive or not. If they are not, then
b1 a1 · · · al−1 and a1 b1 a2 · · · al−1 are connected by a single edge labeled by a1 b1 =
b1 a1 . Now in Γ(w), a and a1 b1 a2 · · · al−1 are connected by a sequence of edges
”lifted” from Γ(a1 w), thus a and b are connected in Γ(w).
Finally, if k = l but a1 and b1 are consecutive, then by E.P. a1 b1 a1 · · · aˆj · · · al−1
for some 1 ≤ j ≤ l − 1 is another reduced word of w. If j = 1, · · · , l − 2 then we
just repeat about argument for the case k < l with b0 = a and a0 = b1 a1 · · · al−1 .
Otherwise if j = l − 1, in particular j > 1, then a1 b1 a1 · · · aˆj · · · al−1 is connected
to b1 a1 b1 · · · aˆj · · · al−1 by an edge labeled by a1 b1 a1 = b1 a1 b1 in Γ(w). Now we
note that b and b1 a1 b1 · · · aˆj · · · al−1 are connected in Γ(w) by lifting a path from
Γ(b1 w), and a and a1 b1 a1 · · · aˆj · · · al−1 are connected in Γ(w) by lifting a path from
Γ(a1 w), we are done.
2. Properties of divided difference operators
Lemma 2.1. If f, g ∈ Z[X] then
∂i (f ∗ g) = (∂i f ) ∗ g + (si . f ) ∗ (∂i g)
Proof.
∂i (f ∗ g)
f ∗ g − si . (f ∗ g)
xi − xi+1
f ∗ g − (si . f ) ∗ g + (si . f ) ∗ g − si . (f ∗ g)
=
xi − xi+1
f ∗ g − (si . f ) ∗ g (si . f ) ∗ g − (si . f ) ∗ (si . g)
+
=
xi − xi+1
xi − xi+1
= (∂i f ) ∗ g + (si . f ) ∗ (∂i g)
=
Theorem 2.2 (Nil-Coxeter relations).
∂i ∂j = ∂j ∂i for |i − j| > 1
∂i ∂i+1 ∂i = ∂i+1 ∂i ∂i+1 for i = 1, · · · , n − 1
∂i2 = 0
Proof. In lecture note 1, Steven and Alex gave a detailed proof.
Definition 2.3. If a1 · · · al is a reduced word of w ∈ Sn , then define ∂w = ∂a1 · · · ∂al .
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Remark 2.4. Because of the fact that Γ(w) is connected (where edges in Γ correspond to only the first two nil-Coxeter relations), ∂w does not depends on the choice
of reduced word, thus is well-defined.
On the other hand, if a1 · · · al is not a reduced word, then ∂a1 · · · ∂al = 0. To see
that we let 1 ≤ j < l be such that u = a1 · · · aj is a reduced word but a1 · · · aj+1
is no longer reduced. Then there is another reduced expression of u that is ended
with aj+1 : b1 · · · aj+1 . Now ∂a1 · · · ∂aj ∂aj+1 ∂al = ∂b1 · · · ∂aj+1 ∂aj+1 · · · ∂al = 0 by
the third relation.
From above consideration, we can conclude
(
∂uv if `(uv) = `(u) + `(v)
(*)
∂u ∂v =
0
otherwise
Lemma 2.5. si ◦ ∂w = ∂w if and only if `(iw) = `(w) − 1
Proof.
si ◦ ∂w = ∂w ⇔ ∂i ∂w = 0 (by definition)
⇔ `(si w) = `(w) − 1 (by *)
Proposition 2.6. If w0 is the longest element of Sn , then
X
∂w0 = a−1
(w)w
δ
w∈Sn
where
Y
aδ =
(xi − xj )
1≤i<j≤n
is the Vandermond determinant, and (w) is the sign of w.
Proof. By definition, for any v ∈ Sn , ∂v can be written of the form
X
∂v =
cv,w w
w∈Sn
P
In particular, we can write ∂w = w∈Sn cw w. By above lemma, we have si ∂w0 =
∂w0 for any i = 1, · · · , n − 1, thus u∂w0 = ∂w0 for any u ∈ Sn . This implies
X
X
X
cw w = u
cw w =
(u. cw )(uw)
w∈Sn
w∈Sn
w∈Sn
Comparing the coefficient, we get cuw = u. cw . Thus if we know the coefficient cw
for some w, then we can derive cw for all w. Indeed, we claim that we know
cw0 = a−1
δ (w0 )
Assume above claim, we note w = ww0 w0 , thus cw = ww0 cw0 = (ww0 )(w0 )a−1
δ =
(w)a−1
.
δ
The only thing left to show is the claim. One reduced expression of w0 is
w0 = (sn−1 · · · s1 )(sn−1 · · · s2 ) · · · (sn−1 )
So,
∂w0 = (∂n−1 · · · ∂1 )(∂n−1 · · · ∂2 ) · · · (∂n−1 )
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We are interested in the coefficient of w0 after ”multiply out” the rhs. For n = 3,
the least non-trivial case we see
1
1
1
∂w0 = ∂2 ∂1 ∂2 =
(1 − s2 )
(1 − s1 )
(1 − s2 )
x2 − x3
x1 − x2
x2 − x3
1
1
1
(−1)3 , claim shown.
clearly cw0 = x2 −x
3 x1 −x2 x2 −x3
The general case can be checked explicitly in a similar fashion.
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